11 THE SPACES L1 AND L2 167
10. Suppose that the measure ju. is bounded and that /x(X) = 1. Let u be a £i-measur-
able mapping of X into itself which leaves invariant the measure p, (Section 13.9,
Problem 24). Then for every function/e J5f1(X, /x) the class U -f=f° u depends only
on the class of/, hence defines an endomorphism of L1, also denoted by /t> U /,
such that NI(£/ /) = N!(/). The restriction of U to L2 is a unitary operator on L2.
(a) If P is the orthogonal projection of L2 on the subspace of vectors which are
invariant under £/, denned in Problem 9, show that N^/)2 <; (/| P /) for all/e L2.
(b) Show that for each/e L1 and each e > 0 there exists an integer n such that, for
all integers m > 0,
(1 m+n-l \
p./_i E £/*/<*.
rt * = m /
(c) Deduce that, for each measurable subset A of X and each e > 0, there exists
an integer n > 0 with the property that for each integer m > 0, there exists an integer
k such that m^k <zm-{- n 1 and p,(A n u~k(AJ) ^ (p,(A))2 e ("Khintchine's
statistical recurrence theorem ").
(d) For each/e 3?^ and each integer n, put
Show that the limit
I m-l
m-*oo m, fcasO
(where P -/denotes a function in the class P /) exists almost everywhere and that
lim Fw dp, = 0. (Use Birkhoffs ergodic theorem.)
(e) Let/e^(X, fM) be such that/(x)^ 0 almost everywhere. Show that, for almost
all x e X, either f(x) = 0 or (P /)(*) > 0. (Consider the set N of points x e X at
which (P /)(*) is either undefined or equal to zero.)
1. Let X be a metrizable compact space and let u: X-*X be a homeomorphism. The
set I of measures ^0 on X of total mass 1 and invariant under u is then a nonempty
vaguely compact subset of M(X) (Section 13.4, Problem 8). A point x e X is said to
be quasi-regular (relative to u) if the sequence of measures
converges vaguely as n -> + °o (necessarily to a measure />tx e I). Let Q denote the set
of quasi-regular points of X. A point x e Q is said to be ergodic (relative to u) if the
measure p,x is ergodic (Section 13.9, Problem 13). Let E be the set of ergodic points
x e Q. A point x e Q is said to be dense ifx belongs to the support of p,x. Let D be the
set of dense points xeQ, The points belonging to R Ğ E n D are said to be regular
(relative to p).
(a) Show that the complement of R (and hence also the complement of Q, E and D)
is negligible with respect to any invariant measure vel. (To show that Q has measure 1
for any measure v e I, apply BirkhofFs ergodic theorem to the functions belonging to a
dense sequence in ^(X), To show that D has measure 1, consider a denumerable basis
(UĞ) for the topology of X, and for each pair (m, n) such that Om c Un, a continuousfore the series with general term aHfn(x) converges absolutely